Applied Mathematics and Scientific Computation

Our research and teaching activities focus on mathematical problems that arise in real-world applications. This involves the mathematical modeling of physical, biological, medical, and social phenomena, as well as the effective use of computing resources for simulation, computation, data analysis, and visualization. Key areas of research in our group are the modeling of bio-films and of materials, computational neuroscience, traffic flow modeling and simulation, the numerical approximation of differential equations, and the solution of large systems of equations. The mathematical modeling of real-world phenomena and the design of modern computational approaches require a broad background in differential equations, fluid dynamics, applied analysis, calculus of variations, functional analysis, probability theory, and other areas.

Faculty

Yury Grabovsky's interests are in the area of Calculus of Variations, PDE, and applications to Continuum Mechanics. He has worked on the mathematical theory of composite materials, non-linear elasticity, and phase transitions in solids. His work on exact relations for effective tensors of polycrystalline composites has combined methods of modern algebra and PDE. His recent work includes a proof of Ball's conjecture in the Calculus of Variations and mathematical analysis of buckling of slender elastic bodies, such as plates rods and shells.

Isaac Klapper is interested mathematical applications in the physical and biological sciences, particularly those with connections to fluid dynamics. While he has worked with mathematical problems arising in solar magnetohydrodynamics, especially some related to dynamo activity, his current focus is on study of mixed physical and ecological issues in microbial communities such as biofilms and microbial mats.

Gillian Queisser works on the interface of numerical methods and scientific computing, high-performance computing, and applications in the life sciences. A particular focus lies on understanding the structure-function interplay in neuroscience, via nonlinear continuum models.

Benjamin Seibold works in Applied and Computational Mathematics, with a specific focus on high-order methods for fluid flows and interface evolution, radiative transfer and kinetic problems, and traffic flow modeling, simulation, and control.

Daniel B. Szyld has worked on many aspects of numerical linear algebra and matrix computations, including eigenvalue problems, sparse matrix techniques, Schwarz preconditioning and domain decomposition, and Krylov subspace methods. He is the editor of several leading journals on applied and numerical linear algebra, and an editor-in-chief of the Electronic Transactions on Numerical Analysis.


Postdoctoral Research Assistant Professors

Dong Zhou works on high-order time-stepping for initial-boundary-value problems, and on finite element methods for incompressible flows, in particular pressure Poisson equation reformulations with electric boundary conditions. He also works on high-order approaches for Hamilton-Jacobi equations, as well as jet schemes.

My research interests include mathematical modeling, dynamical systems, parameter estimation, and partial differential equations. I am particularly interested in models related to the medical field because they present an opportunity to gain insights and make advancements that could profoundly impact humanity.

Parameter estimation for dynamical systems has been the main focus of my recent work. Differential equation models of physical systems often involve parameters whose values are not known and cannot be directly measured. Parameter estimation is a technique that makes use of experimental data to determine values for these unknown coefficients. The problem of estimating parameters is a vital component of model development that arises in almost all applications of mathematical modeling.

Giordano Tierra Chica works on computational fluids dynamics, in particular mixed finite elements methods and diffuse interface methods for multiphase fluid flow simulations, as well as applications of CFD in bio-mathematics.





Current Ph.D. Students

 
Josua Finkelstein Jose Garay Stephan Grein Kathryn Lund-Nguyen Rabie Ali Ramadan Yilin Wu

Former Members

Former Faculty, Postdocs, and Long-Term VisitorsFormer Ph.D. Students
  • 2015: Scott Ladenheim
  • 2014: Stephen Shank
  • 2014: Dong Zhou
  • 2013: Shimao Fan
  • 2012: Meredith Hegg, Kirk Soodhalter
  • 2010: David Fritzsche
  • 2008: Worku Bitew, Xiuhong Du, Abed Elhashash
  • 2007: Mussa Kahssay Abdulkadir, Tadele Mengesha, Kai Zhang
  • 2005: Chao-Bin Liu
  • 2004: Hansun To
  • 2001: Yan Lyansky, Jianjun Xu
  • 2000: Yun Cheng, Judith Vogel, Cheng Wang
  • 1999: Hans Johnston

For more information please consult the department's listing of recent Ph.D. graduates.

Research Profile

Applied MathematicsScientific Computing
  • continuum mechanics and theory of composite materials
  • fluid dynamics and applications
  • modeling and simulation of biological and medical applications
  • modeling and simulation of traffic flow
  • non-linear elasticity and phase transitions
  • neuroscience modeling
  • computational fluid dynamics and fluid-structure interaction
  • high order methods for partial differential equations
  • iterative solution of large linear systems and modern Krylov subspace methods
  • meshfree, particle, and level set methods
  • numerical solution of eigenvalue problems and matrix equations
  • radiative transfer and applications in radiotherapy
  • high-performance computing and supercomputing

Recent Publications by the Group Members

Presentation about the Group

Reflecting the status in November 2010.

Seminar

Special Events

Special Courses

Graduate Program and Courses

In the recent years, several graduate students have completed a Ph.D. or masters degree in the area of Applied Mathematics and Scientific Computing. Information of about the graduate program in Mathematics can be found on the Graduate Program website. Students who are interested in specializing in Applied Mathematics and Scientific Computing can achieve a M.A. in Mathematics with Applied Concentration, as well as a Ph.D in Mathematics, with an advisor in the applied areas. In both cases, students are advised to take (many of) the courses listed below. More detailed syllabi can be found on the Course listing by the Graduate School of the College of Science and Technology. The courses are not taught every semester. Please check the website of the Department of Mathematics for the course schedule.
 

Central Courses

5043. Introduction to Numerical Analysis provides the basis in numerical analysis and fundamental numerical methods.

8007/8008. Introduction to Methods in Applied Mathematics I / II provides the student with the toolbox of an applied mathematician: derivation of PDE, solution methods in special domains, calculus of variations, control theory, dynamical systems, anymptotic analysis, hyperbolic conservation laws.

8013/8014. Numerical Linear Algebra I / II cover modern concepts and methods to solve linear systems and eigenvalue problems.

8023/8024. Numerical Differential Equations I / II present modern methods for the numerical solution of partial differential equations, their analysis, and their practical application.

8107. Mathematical Modeling for Science, Engineering, and Industry. See above for the description of this special course.

9200/9210. Topics in Numerical Analysis I / II are special courses in Numerical Analysis that focus on topics relating to our group members' active research areas. Recent example:

8200/8210. Topics in Applied Mathematics I / II are special courses that are offered by demand. Recent example:

Theoretical Basis

In addition, we recommend courses that provide fundamental theoretical background.

8141/8142. Partial Differential Equations I / II provide a theoretical understanding of many of the equations considered in 8023/8024.

9005. Combinatorial Mathematics relates to many key problems in Scientific Computing, such as mesh generation, load balancing, and multigrid.

9041. Functional Analysis is a theoretical basis for many numerical approximation approaches, such as the finite element method.

9043. Calculus of Variations provides powerful tools for the theoretical study of dynamics, structural mechanics, and material properties.